You may be overcomplicating things, is there any reason you need the stringr package?
df <- data.frame(Date = c("10/9/2009 0:00:00", "10/15/2009 0:00:00"))
as.Date(df$Date, "%m/%d/%Y %H:%M:%S")
[1] "2009-10-09" "2009-10-15"
More generally and if you need the time component as well, use strptime:
strptime(df$Date, "%m/%d/%Y %H:%M:%S")
I'm guessing at what your actual data might look at from the partial results you give.